Inelastic electron scattering on C60 clusters K. · Inelastic electron scattering on C 60 clusters K. Yabana Department of Physics, Niigata University, Niigata 950-21, Japan G. F

Inelastic electron scattering on C60 clusters K. Yabana Department of Physics, Niigata University, Niigata 950-21, Japan

G. F. Bertsch Department of Physics and National Institute for Nuclear Theory, University of Washington, Seattle, Washington 98195

(Received 2 December 1993; accepted 4 January 1994)

We calculate the electronic excitation of C60 by inelastic electron scattering or elect-ron energy loss spectroscopy (EELS). The scattering process is treated in the distorted-wave Born approximation, and the electronic excitations are calculated in a spherical basis model. We find that low energy electrons excite some nonphotoactive modes, in agreement with experiment. Spin triplet modes are poorly excited, even at the lowest electron energies. '

I. INTRODUCTION

The spectroscopy of C60 is very simple for a large mol-ecule, due to its high symmetry. For example, in photoab-sorption measurements, only the excited states with T 1u sym-metry are observed among the ten possible irreducible representations of icosahedral group I h • This has been very helpful in establishing the structure of the molecule and in developing the theory of the electronic excitations. One would like to test the model Hamiltonians with other excita-tions besides those of T1u symmetry, but the experimental information is more difficult to obtain and interpret.

Electron inelastic scattering provides one of the most , useful ways to investigate electronic excitations in general. For low incident energy, there is no strict selection rule and all electronic excited states including the spin triplet states can be excited. The angular distribution of the scattered elec-trons will reflect the symmetry of the excited state.

High resolution electron inelastic scattering data are now available for C60 in the gas phase1,2 and also in thin films,3 using low-energy electrons to excite the molecule. Besides the peaks observed in the photoabsorption measurements, several new features are found in the 2-4 e V region. In this paper, we calculate electron inelastic scattering from C60 , concentrating on the region of the new features. There is also a featUre at high energy loss that has been measured by EELS and interpreted using the plane wave Born approxima-tion for the scattered electron.4 However, low-energy elec-trons are strongly diffracted by the molecule and we shall therefore use the distorted-wave Born approximation (DWBA) for the scattering process. This approximation has been applied in the past to linear diatomic molecules.5 We will also use a spherical basis to expand the wave functions of both the scattering and the bound electrons. This is moti-vated by the high symmetry of the C60 molecule, and it con-siderably simplifies the calculation.

The paper is organized as follows. We present the spherical-basis model for the electronic excitations in Sec. II. The wave function of the scattering electron is strongly dis-torted by the field of the molecule, and these distortion ef-fects are discussed in Sec. m. The scattering is calculated with the full exchange interaction, according to formulas pre-

sented in Sec. IV. The results of the electronic excitations of C60 are presented and compared with experiment in Sec. V.

II. ELECTRONIC EXCITATIONS OF Coo IN A SPHERICAL BASIS MODEL

We have recently applied the spherical basis model to the electronic structure of C60 , calculating the single-particle orbitals and the dipole strength function.6 In this section we shall use the model to describe the particle-hole excitation spectrum.

The electronic Hamiltonian for the 240 valence electrons is taken as

240 p2 240 60 [ -Ze2 ]

2m Iri-Ral +VpsCri-Ra)

240 2 240

+ + vxc[p(ri)]. IJ 1

(2.1)

Here ri(i =1, ... ,240) are the electron coordinates and R a(a=1, ... ,60) represent the eqUilibrium positions of carbon ions. The ion charge number Z equals 4. The potential VpsCr) represents the pseudopotential due to the core electrons. Fi-nally, v xc is an exchange-correlation potentiae depending on the electron density p(rJ

Including only the monopole part of the ion-electron Coulomb potential in Eq. (2.1), we obtain a spherical shell jellium Hamiltonian

240 2 240 2 240 e2

Hjellium= - 60Z + 2: Iri-rj.I' 1=1 ,=1 .> ij

240

+ 2: vxc[p(ri)], (2.2)

where ri> represents larger between ri and R, the radius of C60 • Applying the self-consistent local density approximation to the spherical shell jelIium Hamiltonian of Eq. (2.2), we obtain a spherical self-consistent potential and a correspond-ing spherical density distribution. The self-consistent poten-tial shows a deep valley at the radius of C60 • The single-

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K. Yabana and G. F. Bertsch: Inelastic electron scattering on Cso clusters 5581

10

t1u(7) :::::::::::::=t2u (7) ______ 9u<?)

5 a/6) -9

9(6)

> Q) '-' _hg(6) El 0 t zu( 5) CIl c: -hue?) LU

CIl __ tlg(6)

CI1 t1u(s,LUMO) c.. -5 QJ

'5l hu(s,HOMO) c: Vi ___ hg(4)

gg(4) 9u(3)

-10 -tzu(3) -hgCZ) "llu(l) -ag(O)

-15

FIG. 1. Single particle spectrum in the spherical basis model for 'IT-electron orbitals. The dominant angular momentum for each state is indicated in the parenthesis.

particle orbitals can be labeled by angular momentum and radial quantum numbers n. The states without radial nodes, namely, n =0, correspond to O"-electron orbitals, while the n=1 states correspond to 7T-electron orbitals. Both n=O and n = 1 orbitals are strongly localized near the radius R.

We mixed the states of different angular momentum with the nonspherical electron-ion potential but neglect its effect on the radial wave functions. The eigenstates no longer have the (21 + I)-fold degeneracy of the angular momentum states I but rather the degeneracies of the icosahedral group. Our calculation was not self-consistent in that we did not calcu-late the effect of the icosahedral symmetry-breaking on the electron contribution to the potential. Instead, we parame-tized the strength of the symmetry-breaking interaction, and found it possible to reproduce quite accurately in a spherical basis model the single-particle spectrum obtained with the fully self-consistent local density approximation.6

The single-particle wave function has the following structure:

.ITi u,,(r) "'" ri IfF" (r)=-- C1mY1m(r),

r 1m (2.3)

where the radial wave function is denoted by u,,(r), and cfm are coefficients of the different angular momentum eigen-states (l,m). The r labels the irreducible representation of the icosahedral group, and i distinguishes the degenerate electronic states belonging to the same r. The 7T-7T* transi-tions, that is, transitions among the states with n=l, domi-nate in the low-energy excitation spectrum, and we restrict our attention to these transitions in the following. In Fig. 1 we show the single-particle spectrum of the 7T electrons in

41-_ -

-3 - =:T1U - -> CIl -x

LU --G" z. H. • __ G.

2 '---H9 - T1• -G Tzu

=H G• _9 TZg

_ T1g TZ9

5=0 Even 5=0 Odd 5=1 Even 5=1 Odd

FIG. 2. Particle-hole spectrum in the spherical basis model. All the particle-hole states of the 7T-electron orbitals are included. The electron-electron interaction is weakened by factor 2, to take into account the u-electron screening.

the spherical basis model. The orbitals are bound or are sharp resonances fbr angular momenta up to 1=7; only these orbit-als are included in the diagonalization of the nonspherical interaction. The 1=5 orbitals are split into three multiplets by the nonspherical interaction. The lowest of these, having h" symmetry, is the highest occupied molecular orbital (HOMO). The next higher multiplet derived from the 1=5 set of orbitals is the t I " lowest unoccupied molecular orbital (LUMO). The energies of these states agree well with those calculated using more sophisticated approximations.8

With the single-particle orbitals described above, we construct the particle-hole states taking the hole from the 30 occupied spatial 7T orbitals and the particle from the unoccu-pied 34 spatial 7T orbitals. This yields 510 states of each parity, which are mixed by the residual electron-electron interaction. The residual electron-electron interaction should include the screening effects due to the 0" electrons, since their polarization is not explicitly in the wave function of the configuration interaction calculation. We simply assume here that the residual interaction is the Coulomb interaction screened by factor 2 for all multipolarities. The spectrum obtained by diagonalizing the Hamiltonian matrix is shown in Fig. 2.

Previously the spectrum of particle-bole excitations has been computed in bases utilizing atomic orbitals.9- 11 Our cal-culations agree with these both in the ordering of multiplets within the major groups as well as the larger scale features of the spectrum, except when the multiplets are nearly degen-erate. The excitation energy of the lowest triplet state is in-ferred from various experiments to lie at 1.6 eV, close to our calculated energy of 1.4 eV. Our odd-parity levels fit less well, lying about 0.8 eV lower than experiment. The lowest singlet Tlu state which is observed strongly in photoabsorp-tion experiments has an energy of 3.8 e V, compared to our lowest Ti " at 3.0 eV. The lowest singlet T2u state was as-signed in our previous paperll to a state at 3.0 e V rendered

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5582 K. Yabana and G. F. Bertsch: Inelastic electron scattering on Ceo clusters

visible by vibronic coupling. Its energy is also shifted down in our calculation by about 0.8 eV. However, the difference between the lowest ITlu and the lowest IT2u states, which directly reflects the strength of the residual electron-electron interaction, is reasonable.

III. DISTORTING POTENTIAL FOR Coo

We now turn to the description of electron scattering from C60 • For low-energy incident electrons, say eV, it is essential to include the distortion of the incoming and outgoing electron waves for a quantitative analysis of the angular distributions and cross sections. We deal with this by solving the Schrodinger equation for the ,electrons in the po-tential of the C60 •

We simplify here by considering only the spherically symmetric component of the electron potential. Because of the icosahedral symmetry of C60 , the potential will have components of lnultipolarities L =6,10,12, ... as well. To ar-gue f!Jr the neglect of these, we note that the L =6 compo-nent has only a small effect on the 'T1" orbital. The next mul-tipole, L =10, has little effect on low partial waves up to 1=4. However, as mentioned in the previous section, the HOMO-LUMO orbitals of 'T1" electrons are deriveq from n=1, 1=5 orbitals in the spherical description. We should keep in mind that there might appear several with 1=6,7 character just above the thre;shold, which might strongly affect the scattering. These resopances will be split by the. strong nonspherical interaction, and their detailed po-sitions are nqt easy to predict.

As mentioned in Sec. I the scattering potential should be consistent, with the bound state potential. We use the poten-tial in Eq. (2.2) with three modifications. First, the scattered electron interacts with .240 bound electrons while bound electrons themselves are calculated with the field of the other 239 . electrons only. The se.cond difference is that the

interaction is modified due to the higher momentum of the scattering ele(:tron. We drop the local correlation contribution and, take the exchangy' potential to have the form

e2 Vexch(r)=- 'T1" kF(r) 1- 2k(r)kF(r)

'. k(r)+kF(r) ] x log k(r)-kF(r) , (3.1)

where kF(r) is the local Fermi momentum, defined as usual by kF(r) = [3 'T1"2p(r)] 1/3 , The local momentum of the scat-tered electron, k( r ), is determined by

1i2k(r)2 . 1i2k;(r)2 2m =Ein+ 2m' (3.2)

where 1=2.7 eV is the electron affinity of C60 ,12 and Ein is the incident electron's energy.

There is also long-range contribution to the potential from the polarization of the C60 by the scattering electron, which is not accounted for in the local density approxima-tion. This potential is rather weak and not significant for the spectroscopy, but it can still have an important effect on the

-20

'> ,..... -40 '-'-" >

-60

r [A}

FIG. 3. Spherical distorting potential for the electron at Ee=5 eV.

scattering of low energy electrons. For electron energies be-low 20 e V, we may assume the same polarizability as for static fields. We also consider only the dipole polarizability. The contribution to the potential then has the form

a Vpot(r)=-e2 4' (r>R) r

2 a 2 -e '1>6 r (r<R), ,R (3.3)

with R the 'radius of C60 • We take the value a=96 N for the static polarizability, inferred from the measurement of index of refraction of solid C60 by' the Clausius-Mosotti relation.

We show in Fig. 3 the distorting potential at Einc=5' eV. The potential has a deep valley at the radius of C60 , and is close to the self-consistent spherical potential in the spherical shell jellium model discussed in the previous. section. This potential produces a single-particle state with n =1, I =5 near the threshold. After including the icosahedral perturbation of nonspherical ion-electron interaction, the predicted LUMO of..!1u· symmetry is at about -3 ev' Experimentally, the elec-' tron affinity 'Of C60 ismeasured at 2.7 ev,12 in good agree-ment with the present model.

For higher incident energy where the strongly coupled inelastic channels ine open, an imaginary part of the distort-ing potential should also be included. However, without elas-tic data, it is difficult to make a reliable potential. In the present calculations we do not include the absorptive effect, which would tend to make our our predicted cross sections somewhat high.

IV. DISTORTED WAVE BORN APPROXIMATION

A. Formula for cross sections

In order to assess the relative importance of spin flip and higher multipolarities in the electron scattering, it is impor-tant to calculate the excitation with a full treatment of the exchange interaction between scattering and bound electrons.



We shall use the distorted wave Born approximation, whose basic formulas are given by the following equations (4.4). The differential cross section for the excitation of sin-glet states is given by

( dO') vr ( m ) 2 1 . _ =_ _ I 2 Td1f _ Texchl2 dO, v' 27r1i 2 2 ji ji

i--+J l

and the cross section for triplet state excitation is

(dO') Vf("ni )23' h2 dO, i--+J = Vi' 27r1i2 2' ITfixc I .

(4.1)

(4.2)

In these two equations the direct and the exchange T matri-ces are defined as

d' (_)* (+) e f 2

T/= dr dr' ifJkf (r)ifJk; (r) Ir-r'l pir'),

2 '/'(-)*(r)'/'(+)(r,,),_e_,. P (r' r)' 'l'kf 'l'k; Ir-r'l 'ji ,

(4.3)

where

(4.4)

is the transition density matrix and pir)=Pji(r,r) is the ordi-nary transition density. ' ,

In our model, the transition density for 7r-7r* transitions has the following form:

u(r) u(r') Pa(r,r')=-r-

where u(r)=ul(r) isthe radial wave function of the i or-bital. (n=1). The [ m' are the particle-hole

p p" " amplitudes of the eigenstate ex in the. confiIDIration interac-tion calculation discussed in Sec. II. Our model does not include screening effects of the O' or 7r electrons, This will make the predicted cross sections too large. We thus can only use our model to assess relative cross sections.

To numerically evaluate the scattering wave functions we make a multipole expansion as

(4.6)

The radial wave function F[(k,r) is obtained by numerically solving the radial Schrodinger equa'tion. The T matrix IS now expressed in the following way:

(4.7)

in terms of the partial waves 1 i and 1 J of the and outgoing scattered electron. The coefficient X d1r is given by

where tdir is the radial integral

Similarly, X exch is given by

(4.10)

with the exchange radial integral

The above formulas determine the differential cross .sec-tion for fixed posItions of the ions. Since we d9 not resolve the molecular rotation spectra, we should' add the cross sec-tions including rotational excitation. Assuming the rotational energies much smaller than the electronic excitation energy, the summation over rotational" excitations are equivalent to averaging the cross section over possible orientations of die molecule.

The calculation of Eq. (4.7) is already numerically time consuming, and a further sum over orientations would be beyond our resources. Fortunately, this is not necessary if we average over directions of the incident electron instead. To make this average, we note that the absolute square of the T matrices has the following dependence on the directions of the incident and outgoing electrons:

(4.12)

The average of Eq. (4.12) is made using the fact

(4.13)

where PL(X) is the Legendre polynomial and e represents the relative angle between kr and k;, that is, the scattering angle. The Rn represents the rotation operator for the Euler angles 0,.

Finally, the differential cross section after averaging over orientations is given by


5584 K. Yabana and G. F. Bertsch: Inelastic electron scattering on C60 clusters

and a similar expression is obtained for the triplet excitation. The expression for the angle integrated cross section is much simpler, requiring only a fourfold sum over angular momen-tum indices,

uft= (2:n 2 ) 2 e4(41T)5

X 1 Izxdir vexch 12 £.J 2" ltnlimi-Alrlimi'

Ifmiimi (4.15)

Again, a similar formula is found for the angle-integrated triplet excitation cross section.

B. Plane wave limit

Although the distortion will be strong for C60 , the plane wave limit is useful to understand the qualitative behavior. It will also be useful numerically for the very forward angles where the distortion effect is small and where the conver-gence of the sum over partial waves is poor.

We only discuss the direct term which is dominant at forward angles. The T matrix is given in the plane-wave Born approximation by

d· 41Te2 f . T/(q)=--qr dr e-,qrpft(r) , (4.16)

where q=Jy-k; is the momentum transfer. Using the ex-plicit form of the transition density given by Eq. (4.5), the T matrix may be expressed

Tft(q) = 41T( _i)LPft,LMYLM(q/ dr h(qr)u2(r), LM J

(4.17) where the multipole moments of the transition density Pft,LM are defined by

f A * (A) () u2(r) dr YLM r Pft r =-:y- Pft,LM' r (4.18)

Mter averaging over orientations of the cluster, the differen-tial cross section for the singlet excited state is found to be

av= 41T

{J dr h(qr)u(r)2} 2; I Pji,LMI 2. (4.19)

Since u(r) is peaked at the radius of C60 , the radial integral is approximately given by h(qR). For low momentum transfer, only multipolarities with small L are important. The

(4.14)

strong transitions will then be to states which contain low L. This depends on the r of the icosahedral group; the lowest Lof different representations are given in Table 1. The strongest states at forward angles should be states with T 1 u and H g symmetry, since they can carry L = 1 and 2. There are no low electronic excitations of A g symme-try, which would also be strong by this argument. This ap-proximate selection rule will be useful to understand the dis-torted wave results and their comparison with measurements.

V. RESULTS AND DISCUSSION

EELS data at low bombarding energyl-3 show some structures in the low excitation region which are absent in the photo absorption measurements. These include peaks at 2.2 and at 3.0 e V, below the dipole excitations starting at 3.8 e V. There is also a weak feature observed at 1.5 e V in the thin film data.

We will attempt an assignment of the electronic excita-tions for the observed peaks, making use of the DWBA cal-culations. The gas phase data has the energy of outgoing electrons fixed at 3 eV, and we will calculate the DWBA cross sections under these conditions. We set the excitation energy of the low lying states to the average value, 2 e V for the even parity and 3.5 eV for the odd parity excitations. The incident electron energies are then 5 e V for even parity and 6.5 e V for odd parity excitations. We include the partial waves of the incoming and outgoing waves up to 1=12,. much larger than the classical grazing angular momentum kiR -4.6 .. However, as will be seen, this cutoff on lis not sufficient for convergence of the low multipole excitation cross sections at forward angles.

In Table I1(a), we show the integrated cross sections for the low lying excitations. The states are the lowest of each symmetry, and are marked in the excitation spectrum of Fig. 2. From the table it may be seen that the integrated cross sections exciting singlet states depend strongly on the par-

TABLE 1. Lowest multipolarity L in the representations r of the icosahedral group.

r L

Ag 0 T lu 1 Hg 2 T2u 3 Gu 3 Gg 4 Hu 5 T 1g 6 T2g 8 Au


K. Yabana and G. F. Bertsch: Inelastic electron scattering. on Ceo clusters 5585

TABLE II. The integrated cross sections (a) and the forward angle differential cross sections (b) for the electron inelastic scatterings to low lying electronic excitations. Final electron energy is set to eV.

(a) Integrated cross sections (A2) S=O S=l

Even Odd Even Odd

T2g 0.485 Gu 0.605 T2g 3.179 T2u 2.457 Tig 0.519 T2u 1.434 T ig 2.151 Tlu 2.778 Gg 1.105 Hu 0.709 Hg 3.617 Gu 2.763 Hg 19.748 T lu 15.237 Gg 2.318 Hu 2.319

(b) Forward angle differential cross sections (Az/Sr) S=O

Even Odd

T2g 0.009 Gu 0.074 T ig 0.003 T2u 0.332 Gg 0.055 Hu 0.039 Hg 8.414 T lu 41.358

ticular state involved, but the cross sections are nearly the same for different triplet excitations. This follows from the character of the interaction. For triplet excitations, only the exchange interaction contributes. It requires high momentum transfer and ·so does not exhibit particular selectivity with respect to the spatial symmetry of the excitation. The singlet excitations are dominated by the direct interaction for low momentum transfer, and this produces very large cross sec-tions for low multipolarities. As we discussed in Sec. IV B the cross section for the direct process depends on the mo-mentum transfer as q-4[h(qR)f with R the radius of C60 , in the plane wave limit. It appears that the direct process dominates for singlet excitations of states containing L and the exchange process is as important for other states.

The cross sections for exchange scattering are. much smaller than the low-L singlet excitation cross sections, even at the lowest incident energy. There does not seem to be any preference among states of different icosahedral symmetry, reflecting the large momentum transfer. All the triplet exci-tations are excited nearly equally, though the forward cross sections show some selectivity for the low multipole excita-tions.

A. Assignment for the observed peaks

It seems likely that the 1Tlu state should be assigned to the observed photoabsorption peak at 3.8 e V. The only other low state that has a substantial cross section in our model is the IH g state. The predicted excitation energy of this state is about 2 eV in our model and 2.5 eV in Ref. 9. Thus we feel confident that the observed peak at 2.2 e V should be as-signed to the Hg configuration. (Note however that Ref. 10 predicts the lowest IHg at 3.15 eV.) Quantitatively, the ex-periment shows stronger excitation of the 2.2 e V peak than the dipole-allowed peak at 3.8 e v.l

Table II(b) displays the differential cross sections in the forward direction. Although our calculation predicts a larger integrated cross section for the IHg than the IT1u state, the H g is weaker in the forward direction. Two deficiencies of our calculation may be responsible. First, as will be dis-

S=1

Even Odd

T2g 0.085 T2u 0.937 T ig 0.043 flu 2.692 Hg 0:319 G u 0.338 Gg 0.121 Hu 0.213

cussed, the partial wave expansion is not fully converged in our calculation for forward scattering. Second, the long-range Coulomb interaction should be screened by the polar-ization of the C60 • Screening would be stronger for the lower multipole state and reduce more efficiently the 1T lu excita-tion than IH g excitation.

We cilnnot assign definitely the structure at 3 e V, but there are several possibilities. In view of the energy spectrum of Fig. 2, the excitation is probably one of the odd parity states. The integrated cross sections are nearly the same ex-citing the IT 2u state or one of the triplet states. At the for-ward angles relevant to the EELS measurments, the 3TIu state has the largest differential cross section, and so this assignment would be preferred.

On the other hand, in the optical absorption measure-ment, two clear peaks are observed around 3.1 eVY We pre-viously made an analysis in which we concluded that these states are vibrational modes built on the IT 2u electronic excitation.ll Therefore, it is also quite plausible to assign the peak at 3.0 e V as the IT 2u- The fact that the peak at 3 e V is very sharp from the EELS data on the gas phase,1 in contrast to the broad features of 2.2 and 3.8 e V, is also consistent with this interpretation, since the optical absorption peaks at 3.1 eVare also quite sharp.9 At this point, it is difficult to make a definite assignment selecting between the two possibilities, the IT 2u and the 3T lu excitations.

B. Angular distribution

Though the measurements of the angular distributions of the scattered electrons are not available yet, they reflect the excitation structures and are useful for the spectroscopic studies. We present some calculations of the ·angular distri-bution.

In Fig. 4, we show angular distributions of the three singlet excitations, IT Iu, IH g, and 1:[ 2u. They are character-ized by their lowest allowed Qlultipoles, which are L = 1,2,3, respectively. The differential cross sections are forward


5586 K. Yabana and G. F. Bertsch: Inelastic electron scattering on Cso clusters

....... .... V) ........

N 0« 40 '--' c o . .j:j u Q)

V) (/) (/)

e 20 u ell . .j:j C

o

Singlet Excitations

x 10

--T,u ._ ......... _ .. H .

9 -- - _. T 2u

50 100 150 Angle (Degree)

FIG. 4. Angular distribution for the singlet excitations IT lu' IH g' and IT 2u, calculated by the DWBA. The outgoing electron energy is set to 3 eV.,

peaked, with the' smaller angular range for the lower multi-pole excitations. '

Figure 5 shows the., comparison of the distorted wave Born and the plane wave Born approximations for the IH excitations. The plane wave Born results are calculated two ways: the formula of Sec. IV B which include all the partial waves of the incoming and outgoing waves, and the DwBA code without the distorting potential. The compari-son of the two plane wave calculations thus shows the con-vergence of the calculation with respect to the partial wave expansion up to l = 12.

At very forward angles, the plane wave calculation with partial wave cutoff underestimates the cross section by 30%. The DWBA calculation may underestimate the same amount of the cross section at forward direction. Though the angular

Hg Singlet Excitaiton

, . DWBA(lmax=12)

_ ........... PWBA , --_. PWBA(I =12) :\. max .

50, 100 150 Angle (Degree)

FIG. 5. Angular distribution forthe IHg excitation, comparing' the DWBA and the PWBA. In the PWBA calculation, the calculations are shown with and without the partial wave expansion of the scattered electron.'

'3 ....... .... V) ........ Triplet Excitations

N 0« --....... ·c 0 2 . .j:j u --TJu Q)

U) - - - - T 2u (/) (/)

..... -......... Hg 0 .... u

roo • .j:j c Q) .... / Q) "-"-(5

............ i ..... : ..........•............................ .......... " 0

0 50 100 1.50 Angle (Degree)

FIG. 6. Angular distribution for the triplet excitations 3T lu, 3T 2u, and 3H g' calculated by the DWBA. The outgoing electron energy is set to 3 eV.

distributions show difference, the integrated cross sections coincide to an accuracy within 1%. The difference between two plane wave calculations is largest for the IH g state in the present calculation.

Compared with the plane wave result, the DWBA cross section is reduced significantly, showing the importance of the distottion effect for the quantitative analysis.

In Fig. 6, the angular distributions for some triplet exci-tations are shown. The angular distribution is flat for many excitations, reflecting various multipole contribution to aver-age the cross section. The 3T1u excitation is exceptional, showing a forward peaked angular distribution. The 3T 2u ex-citation included in the figure is the lowest electronic excita-tion, which may be responsible for the 1.5 eV structure ob-served in the thin film data.3

VI. SUMMARY

We developed a theoretical framework for interpreting EELS measurements on C60 • We rely on a spherical basis for constructing wave functions both for the electronic excita-tions and the scattering electrons.

Besides the dipole allowed excitations, a singlet excita-tion with Hg symmetry, which includes L ""=2+ component, is found to be excited strongly, and is assigned to the ob-served structure around 2.2 e V in the EELS measurement.

The singlet excitations in the low incident energy follow the formula in the long wavelength limit of the plane wave approximation, where the cross section for the excitation of multipole L is proportional to q-4[h(qR)]2 with R the ra-dius of C60 • The triplet excitations are weak and do not show the selectivity of the final excitations, reflecting the large momentum transfer associated with the exchange process.



ACKNOWLEDGMENTS

One of the authof§(KY.) is grateful for the hospitality of INT, Univ. of Washington, where this work was initiated. Financial support was provided by the Department of Energy under Grant No. DE-FG06-90ER40561. . .

IC. Bulliard, M. Allan, and S. Leach, Chern. Phys. Lett. 209, 434 (1993). 2 A. Burase, T. Dresch, and A. Ding, Z. Phys. D 26, (1993), S294. 3 A. Lucas, G. Gensterblurn, J. J. Pireaux, P. A. Thiry, R. Caudano, J. P ..

Vigneran, Ph. Larnbin, and W. Kratschrner, Phys. Rev. B 45, 13694 (1992).

4 A. Bulgac and N. Ju, Phys. Rev. B 46, 4297 (1992). 5 A. FIifIet and V. McKoy, Phys. Rev. A 21, 1863 (1980). 6K. Yabana and G. F. Bertsch,"Phys. Scripta (in press).' 7 O. Gunnarsson and B. Lundqvist, Phys. Rev. B 13, 4274 (1976). 8 J. W. Mintmire, B. I. Dunlap, D. W. Brenner, R. C. Mowrey, and C. L

White, Phys. Rev. B 43, 1428i (1991); B. 1. Dunlap, D. W. Brenner, J. W. Mintmire, R. C. Mowrey, and C. T. White, J. Phys. Chern. 95, 5763 (1991); S. Saito and A. Oshiyarna, Phys: Rev. Lett. 66, 2637 (1991); J. L.

Martins, N. Troullier, and J. H. Weaver, Chern. Phys. Lett. _ 180, 457 (1991); J. H. Weaver,J. L. Martins, T. Korneda, Y. Chen, T. R. Ohno, G. H. Kroil, N. Troum;;r, R. E. Haufler, and R. E. Srnalley, ibid: 66, 1741 (1991); H. P. Liithi and J. Alrnliif, Chern. Phys. Lett. 135, 357 (1987); P. W. Fowler, P. Lazzeretti, and R. Zanasi, ibid. 165, 79 (1990); W. Y. Ching, M.-Z. Huang, Y.-N. Xu,w. G. Hacier, and F. T. Chan, Phys. Rev. Lett. 67, 2045 (1991); L. Ye, A. J. Freernan, and B. Delley, Chern. Phys. 160,415 (1992); V. de Coulon, J. L. Martins, 'lind F. Reuse, Phys. Rev. B 45, 13671 (1992). .

9 S:Larsson, A. Volosov, and A. Chern. Phys. Leit. 133, 501 (1987). 10F. Negri, G. Orlandi, and F. Zerbeito,Chern. Phys. Lett. 144, 31 (1988). 11 K. Yabana and G. F. Bertsch, Chern. Phys. Lett. 197, 32 (1992). __ 12S. H. Yang, C. L. 1.'ettiette, 1. O. Cheshnovsky, and R. E.

Smalley, Chern. Phys. Lett. 233 (1987). 13 R. L. Whetten, M. M. Alvarez, S. J.Anz, K. E. Schriver, R. D. Beck, F. N.

Diederich, Y. Rubin, R. Ett!, C. S. Foote, A. P. Darrnanyan, and J. W. Arbogast, Mat. Res. Soc.Syrnp. Prac. 21)6, 639 (1991); Z. Gasyna, P. N. Schatz, J. P. Hare, T. J. Dennis, H. W. Kroto, P. Taylor, and D. R. M. Walton, Chern. Phys. Lett. 183, 283 (1991); R. E. HaufIer, Y. Chai, L. P. F.

. Chibante, M: R. F'raelich, R. B. Weisrniln, R. F. Curl, and R. E. Smalley, J. Chern. 95, 2197 (1991).


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Inelastic electron scattering on C60 clusters K. · Inelastic electron scattering on C 60 clusters K. Yabana Department of Physics, Niigata University, Niigata 950-21, Japan G. F